List chromatic number conjecture for multipartite uniform hypergraphs

Let $k\geqslant 2$, let $\Delta$ be sufficiently large, and let $H$ be a $k$-partite $k$-graph of maximum degree at most $\Delta$. Write $\chi_\ell(H)$ for the list chromatic number of $H$. List chromatic number conjecture for $k$-partite $k$-graphs. There is a constant $c=c(k)>0$ such that

$$\chi_\ell(H)\leqslant c\log\Delta.$$

This extends the Alon–Krivelevich conjecture from bipartite graphs to $k$-partite $k$-graphs and is supported by results for random and complete multipartite hypergraphs. The case $k=2$ matches the cited bipartite conjecture; the general case remains open.